Chap. X] SuM AND NuMBER OF DiVISORS. 283 
account of the order of the factors. The number of pairs of relatively prime 
integers ^, 17 for which ^r^^n is therefore 
y=i 
For the preceding C and r(n), it is proved that 
r(n)=S^,/.[P], . t = [V^], 
^in)=^(log.n+'-^+2C-l)+m, " C- I H^, 
IT IT 8=2 S 
where m is of the order of magnitude of n\ 8>y/2, while 7 is determined by 
2)s~^ = l (s = 2 to 00). Moreover, T(n) is the number of pairs of integers 
X, y for which xy^n. He noted that 
(7(l)+(r(2) + ...+(r(n)=Ssr^l 
8=1 Ls-i 
and that the difference between this sum and ir^n^/12 is of an order of magni- 
tude not exceeding n loge n. 
G. H. Burhenne^^ proved by use of infinite series that 
r(n)=i2)/"K0), fix)^- "^^ 
and then expressed the result as a trigonometric series. 
V. Bouniakowsky^^ changed x into x^ in (6), multiplied the result by x'^ 
and obtained 
(x^ +x' +x^ + . . .)* = x*+(r(3)x'2_^ . . . +(r(2w+l)a:^"'+H .... 
Thus every number 8m+4 is a sum of four odd squares in (r(2w+l) ways. 
By comparing coefficients in the logarithmic derivative, we get 
(8) (l2-2m+l)(r(2m+l) + (3^-2m-l)(7(2m-l) + (52-2m-5)(r(2m-5) 
+ ...=0, 
in which the successive differences of the arguments of <r are 2, 4, 6, 8, ... . 
For any integer N, 
(9) {l^-N)a{N) + {S^-N-h2)(r{N-l-2) + i5''-N-2-3)a{N-2'3) 
+ -..=0, 
where o-(O) , if it occurs, means A^/6. It is proved (p. 269) by use of Jacobi's^° 
result for s^ that 
l+x+x'+x'+ . . . =P^= (i+x)a+x'){l+x') . . . 
{l-x')(l-x'){l-x')..., 
"Archiv Math. Phys., 19, 1852, 442-9. 
»M6m. Ac. Sc. St. P^tersbourg (Sc. Math. Phys.), (6), 4, 1850, 259-295 (presented, 1848). 
Extract in Bulletin, 7, 170 and 15, 1857, 267-9. 
